Temporal dissipative solitons in the Morris-Lecar model with time-delayed feedback

We study the dynamics and bifurcations of temporal dissipative solitons in an excitable system under time-delayed feedback. As a prototypical model displaying different types of excitability, we use the Morris-Lecar model. In the limit of large delay, soliton like solutions of delay-differential equations can be treated as homoclinic solutions of an equation with an advanced argument. Based on this, we use concepts of classical homoclinic bifurcation theory to study different types of pulse solutions and to explain their dependence on the system parameters. In particular, we show how a homoclinic orbit flip of a single-pulse soliton leads to the destabilization of equidistant multi-pulse solutions and to the emergence of stable pulse packages. It turns out that this transition is induced by a heteroclinic orbit flip in the system without feedback, which is related to the excitability properties of the Morris-Lecar model

On the stability of periodic orbits in delay equations with large delay

We prove a necessary and sufficient criterion for the exponential stability of periodic solutions of delay differential equations with large delay. We show that for sufficiently large delay the Floquet spectrum near criticality is characterized by a set of curves, which we call asymptotic continuous spectrum, that is independent on the delay.Comment: postprint versio

Temporal dissipative solitons in time-delay feedback systems

This is the final version. Available from American Physical Society via the DOI in this record.Localized states are a universal phenomenon observed in spatially distributed dissipative nonlinear systems. Known as dissipative solitons, auto-solitons, spot or pulse solutions, these states play an important role in data transmission using optical pulses, neural signal propagation, and other processes. While this phenomenon was thoroughly studied in spatially extended systems, temporally localized states are gaining attention only recently, driven primarily by applications from fiber or semiconductor lasers. Here we present a theory for temporal dissipative solitons (TDS) in systems with time-delayed feedback. In particular, we derive a system with an advanced argument, which determines the profile of the TDS. We also provide a complete classification of the spectrum of TDS into interface and pseudo-continuous spectrum. We illustrate our theory with two examples: a generic delayed phase oscillator, which is a reduced model for an injected laser with feedback, and the FitzHugh-Nagumo neuron with delayed feedback. Finally, we discuss possible destabilization mechanisms of TDS and show an example where the TDS delocalizes and its pseudo-continuous spectrum develops a modulational instability.Engineering and Physical Science Research Council (EPSRC)European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreemen

Pedestal and Er profile evolution during an edge localized mode cycle at ASDEX Upgrade

The upgrade of the edge charge exchange recombination spectroscopy diagnostic at ASDEX Upgrade has enabled highly spatially resolved me asurements of the impurity ion dynamics during an edge-localized mode cycle ( ELM ) with unprecedented temp oral resolution, i.e. 65 μ s. The increase of transport during an ELM induces a relaxation of the ion, electron edge gradients in impurity density and fl ows. Detailed characterization of the recovery of the edge temperature gradients reveals a difference in the ion and electron channe l: the maximum ion temperature gradient T i is re-established on similar timescales as n e , which is faster than the recovery of T e .Afterthe clamping of the maximum gradient, T i and T e at the pedestal top continue to rise up to the next ELM while n e stays constant which means that the temperatur e pedestal and the resu lting pedestal pressure widen until the next ELM. The edge radial electric fi eld E r at the ELM crash is found to reduce to typical L-mode values and its ma ximum recovers to its pre-ELM conditions on a similar time scale as for n e and T i . Within the uncertainties, the measurements of E r align with their neoclassical predictions E r,neo for most of the ELM cycle, thus indicating that E r is dominated by collisional processes. However, between 2 and 4 ms af ter the ELM crash, other contributions to E B ́ fl ow, e.g. zonal fl ows or ion orbit effects, could not be excluded within the uncertainties.European Commission (EUROfusion 633053